Linear analogue of additive theorems · the additive theory have been obtained, and many questions are wide open. The goal of this master thesis is to understand what has been established

UNIVERSITY OF BORDEAUX-UNIVERSITY OF

PADOVA

MASTER THESIS

Linear analogue of additive theorems

Author:Carnot DONDJIO KENFACK

Supervisor:Pr. Christine BACHOC

A thesis submitted in fulfillment of the requirementsfor the degree of Master

in the

ALGANTInstitut de mathématiques de Bordeaux

July 24, 2017

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Declaration of AuthorshipI, Carnot DONDJIO KENFACK, declare that this thesis titled, “Linear analogue of ad-ditive theorems” and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research de-gree at this University.

• Where any part of this thesis has previously been submitted for a degree orany other qualification at this University or any other institution, this has beenclearly stated.

• Where I have consulted the published work of others, this is always clearlyattributed.

• Where I have quoted from the work of others, the source is always given. Withthe exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I havemade clear exactly what was done by others and what I have contributed my-self.

Signed:

Date:

iv

University of Bordeaux-University of Padova

AbstractBordeaux 1

Institut de mathématiques de Bordeaux

Master

Linear analogue of additive theorems

by Carnot DONDJIO KENFACK

In additive number theory, the sets A + A = {a + a : (a, a) ∈ A2}, where A is afinite subset of an abelian group (G,+) are the objects of study. In particular, one isinterested in the structure of A when A + A is small. For example, A is a coset of asubgroup of G, if and only if A + A has the same cardinal as A. But what happenswhen A+A is slightly larger ? the case of G = Z is well understood, since Freiman’sand Ruzsa’s works, and involves multidimensional arithmetic progressions. In thismaster thesis we study the transposition of these questions to a linear setting estab-lished by C. Bachoc, O. Serra and G. Zémor in [1]. Here the group is replaced by anextension of fields, the finite subsets by finite dimensional linear subspaces, and theaddition by the multiplication. In this framework, only a small piece of the classicalresults of the additive theory have been obtained.


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AcknowledgementsFirst of all, I will start by thanking God for the blessing he gave me during all times inmy life. I will take the opportunity to say many thanks for my father Fidéle Dondjio,my mother Marie Christine Dondjio, Tsomeza Denis and my brothers for all theirsupport.

I would also express my sincere gratitude and appreciation to my supervisorProf. Christine Bachoc for his enormous support, comments, suggestions, advice,and encouragement at every stage of this essay.

My special thanks Prof. Marco Garuti for his advice and fatherly care .I offer my regards and appreciation to all the ALGANT staff.

I am very delighted to thank all the lecturers who have substantially boosted myknowledge .

My special thanks go to my friend Mathias Agbor and all my friends for theirvaluable encouragements and help.

Finally, I would like to express my deepest love to ALGANT students 2015/17.You guys made my stay at ALGANT amazing. I love you all. . . .

vi

Contents

Declaration of Authorship iii

Abstract iv

Acknowledgements v

1 Introduction 1

2 Preliminaries and special case of Vosper’s theorem 42.1 Basic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Connectivity in field extension . . . . . . . . . . . . . . . . . . . . . . . 52.3 The 2 atome of Vosper space . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Connection between Quadratic form and Vosper’s theorem and Applica-tion to finite field 123.1 Motivation, Sidon spaces and Quadratic forms . . . . . . . . . . . . . . 12

3.1.3 Sidon spaces and quadratic form . . . . . . . . . . . . . . . . . . 133.2 Codes in the space of quadratic forms over a field . . . . . . . . . . . . 15

3.2.1 Action ofMn(F ) on quadratic forms . . . . . . . . . . . . . . . 153.3 Vosper’s theorem for finite field . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.7 The equations satisfied by {Xt, t ∈ Tn} . . . . . . . . . . . . . . . 193.3.12 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Vosper’s theorem for fields with valuations 274.1 Motivation and Main result . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Another idea when L has no F− valued place . . . . . . . . . . . . . . 29

5 Conclusion 31

Bibliography 32

vii

I Dedicate to my Big family AZONSOP in Santchou, Douala,Libreville and those from AKO.. . .

1

Chapter 1

Introduction

Additive number theory, is a speciality of number theory, which studies subsets ofintegers and their behaviour under addition. More abstractly, the field of "additivenumber theory" includes the study of abelian groups and commutative semigroupswith an operation of addition. Additive number theory has close ties to combinato-rial number theory and the geometry of numbers. Two principal objects of study arethe sumset of two subsets A and B of elements from an abelian group G,

A+B = {a+ b : a ∈ A, b ∈ B}

and the h−fold sumset of A,hA = A+ · · ·+A︸︷︷︸

h times. However, there are two type of problems in this subject. The first one iscalled direct problem. In the direct problem, we start with two sets A and B and tryto deduce an information of A+B, or we can start by A and determine the structureof hA (h is a positive integer). For example, determining which elements can berepresented as a sum from hA, where A is a fixed subset.

The second one is called inverse problems, often over more general groups thanjust the integers, that is, given some information about the sumset A+B, the aim isto find information about the structure of the individual sets A and B.

Unlike problems related to classical bases, as described above, this sub-area oftendeals with finite subsets rather than infinite ones. A typical question is to know whatis the structure of a pair of subsets whose sumset has small cardinality (in relation to|A| and |B|). In the case of the integers, the classical Freiman’s theorem (1964, 1966)provides a potent partial answer to this question in terms of multi-dimensional arith-metic progressions.

Another typical problem, is simply to find a lower bound for |A + B| in termsof |A| and |B| (this can be view as an inverse problem with the given informationfor A+B being that, |A+B| is sufficiently small and the structural conclusion thenbeing that either A or B is the empty set, such problems are often considered directproblems as well). Examples the Cauchy–Davenport Theorem, which is the oldestresult states that for a given subset S and T of a group of prime order p then, |S+T | ≥min{p, |S| + |T | − 1}. The Vosper’s theorem (1956) in [14], characterizes the sets forwhich the equality holds.

After the first version of Vosper’s theorem due by Vosper which states that,

Theorem 1.0.1. (Vosper) Let S, T be a subsets of a group of prime order p such that |S| ≥2, |T |, and |S + T | < p− 1. Then either |S + T | ≥ |S|+ |T |, or S and T are in arithmeticprogression with same difference.

The improvement became an important problem and then, G. A. Freiman (1961)improves what has been done by Vosper by taking S = T .

2 Chapter 1. Introduction

Theorem 1.0.2 (Freiman). Let S be a subset of a group of prime order p such that |S| <p/35. Suppose |S+S| < 2|S|+mwithm ≤ 2

5 |S|−3. Then, S is contained in an arithmeticprogression of length at most |S|+m+ 1.

In another direction direction the Cauchy-Davenport has been generalized byMann in [9] for any abelian group. And in (2000) O. Serra and G. Zémor worktogether on the generalization of Vosper theorem. The statement is given by

Theorem 1.0.3 (Serra, Zémor). Let S, T be a subsets of a group of prime order p. If |S +

T | ≤ min{p − 2, |S| + |T | + m}, 2 ≤ |S|, |T |, and |S| ≤ p −(m+ 4

2

). Then, S is a

union of at most m+ 2 arithmetic progression with same difference.

Concerning the Vosper’s theorem, we can ask ourself some natural questions.

(1) Is it possible to translate all this result in the case field extension?

(2) Is it possible to transpose the Vosper’s theorem in any field extension?

For the first question the answer is yes for instances the linear analogue Cauchy -Davenport was proved by Eliahou and Lecouvey (2009) in [3].

Theorem 1.0.4 (Cauchy-Davenport). LetL/F be an extension such thatF is algebraicallyclosed in L. For every pair of subspaces S, T ⊂L of finite dimension over F,

dimST ≥ min {dimL, dimS + dimT − 1} .

The second question is solve in part, and since (2015) C. Bachoc, O. Serra andG. Zémor try to give an answer to this question in [1]. The transposition of thisproblem to linear setting is under study. The group is replaced by an extension offields, the finite subsets by finite dimensional linear subspaces, and the addition bythe multiplication. In this framework, only a small piece of the classical results ofthe additive theory have been obtained, and many questions are wide open.

The goal of this master thesis is to understand what has been established by C.Bachoc, O. Serra and G. Zémor in [1], and try to improve for other field like the fieldof power Laurent series K((t)) where K is a algebraically closed field.

Our main result is:

Theorem 1.0.5. Let F a finite field L/F be a finite extension of prime degree such that Fis algebraically closed in L. Let S, T be a subspaces of L such that 2 ≤ dimS, dimT anddimST ≤ dimL − 2. If dimST = dimS + dimT − 1, then there are bases of S andof T respectively of the form

{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1

}for some

g, g′, a ∈ L.

To reach our objective, we are going to organize our work in three chapters andconclusion.

In the Chapter 2, we considerL a finite extension of F with F algebraically closedin L. In the first part of this chapter, we fix the preliminary materials, define thenotion of k−fragment and k−atom. In the second part, we are going to keep ourattention on the 2−atom and we will see that 2− atom has the dimension 2 if wesuppose that [L : F ]− 2 is prime. under this hypothesis, we will deduce our specialcase of Vosper’s theorem.

In the Chapter 3, we will first motivate why is important to study the fields likealgebraically closed field and finite field. By showing that, in the case of anisotropicquadratic form of at least three variables there is an extension which does not satisfy

Chapter 1. Introduction 3

Vosper. Secondly, we will give the connection between quadratic form and Vosper’stheorem. To deal with that, we define two notions, the notion of Sidon space and theweight of quadratic form. In the first time, we make an equivalence with Sidon spaceand weight of quadratic form. As, the idea is to show that if L/F is field extensionand A a 2−atom of S where S is a subspace of L of dimension greater or equals to2. Then, the dimension of A is 2. In the second time, we will redefine our probleminto space of quadratic forms. The third point of this chapter, is to consider the finitefield F = Fq with q = pm where m is a non-negative integer and p a prime numberand L/F the finite extension of prime degree such that F is algebraically closed inL. And apply what we have done to deduce the results.

The last Chapter consists to prove Theorem 4.1.2 when L is a transcendentalextension. We start by explaining why it is possible to think about the transcendentalcase. Our motivation come from [12]. It will turn out to be true, but only in part. Thefact that the transcendental extensions have a valuation rings will be a key point,because we should transfer the structure of spaces with small products to sets withsmall sumsets in totally ordered abelian groups. We will first prove our main result,which consists to take L/F field extension with valuation such that, the residue fieldLv of L is contained in F.And show that under this hypothesis the Vosper’s theoremholds. The second point, it is to identify these fields satisfying this hypothesis andapply the main result.

4

Chapter 2

Preliminaries and special case ofVosper’s theorem

The goal of this chapter, is to introduce some basic materials useful to establish theprincipal results of this thesis and prove the Vosper’s theorem in the special case.Let F a field. Let L an extension of F, we suppose F is algebraically closed in L.We are going to give two important lemmas useful to get all the result of this work.In second point, we are going to introduce as in [1] the notion of k−fragment andk−atom. We will give some properties of k−fragment, and we will focus our atten-tion on the 2−atom and then, we will give the properties of 2−atoms of S, whereS is a subspace of L. It will come out that, under the condition that [L : F ] − 2 is aprime, the 2−atom has dimension 2. Therefore, it will follow the prove of Vosper’stheorem using the Lemma 2.1.2 and Lemma 2.1.3. [10], [11]

2.1 Basic materials

In this section, we are giving some basic materials useful to get the results of thischapter. We will start by recalling the Cauchy-Davenport inequality and then, wewill state and prove two important lemmas.

Theorem 2.1.1 (Cauchy-Davenport). Let L/F be an extension such that F is algebraicallyclosed in L. For every pair of subspaces S, T ⊂L of finite dimension over F,

dimST ≥ min {dimL, dimS + dimT − 1} .

Proof. See [3]

Lemma 2.1.2. Let L be an extension of F such that F is algebraically closed in L. Supposethat S is F-subspace of dimension s of L, and that A is a subspace of dimension 2 generatedby {1, a} such that dimAS = dimS + 1 ≤ dimL − 1. Then there exists g ∈ S such that{g, ga, · · · , gas−1} is basis for S.

Proof. Since A is a subspace of dimension 2 generated by {1, a}, it follows that AS =S + aS and then

dim(AS) = dim(S + aS) = dimS + dim(aS)− dim(S ∩ aS)

= 2dimS − dim(S ∩ aS).

Hence, dim(S ∩ aS) = 2dimS− dimAS = 2dimAS− dimS− 1 = dimS− 1. To showthe existence of geometrical basis, we will proceed by induction on the dimensionof S. Let us suppose the case that, dimS = 2 and g′ the generator of S ∩ aS meaning

2.2. Connectivity in field extension 5

that S ∩ aS = Fg′. Then, g′ = ag with g ∈ S. We claim that, g does not belongsin aS. If g ∈ aS then, g ∈ S ∩ aS. Therefore, there exist some λ ∈ F ∗ such thatg = λg′ = λag. It is equivalent to say that, a ∈ F contradiction. So, {g, ag} is a basisof S. Suppose that, dimS = s ≥ 2 and let S′ = S ∩ aS. Then, dimS′ = s− 1,moreoverS′ also satisfies dimAS′ = dimS′ + 1. Indeed, we have dimS′ + 1 ≤ dimAS′ thisinequality comes from Theorem 2.1.1. Since, AS′ = S′ + aS′ and S′ = S ∩ aS itfollows that, S′ ⊆ aS.And then,AS′ = S′+aS′ ⊆ aS. Therefore, we have dimS′+1 ≤dimAS′ ≤ dimaS = dimaS = dimS′ + 1. By induction hypothesis, S′ has a basis ofthe form {g′, g′a, · · · , g′as−2}. Since g′ ∈ aS, for some g ∈ S, g′ = ag. Moreover gis not in S. In fact, if g ∈ S′ it follows that, g =

∑s−1i=2 λig

′ai + λ1g′. The fact that,

g′ = ag implies that g =∑s−1

i=2 λigai+1 + λ1ag for some λi ∈ F. This would mean

that, degF ≤ s− 1 < dimL contradiction. So, {g, ga, · · · , gas−1} is a basis of S.

Lemma 2.1.3. Let L be an extension of F such that F is algebraically closed in L. Let S, Tthe F-subspace of L with s = dimS, t = dimT ≥ 2. Suppose that there exists a basis of S ofthe form {g, ga, · · · , gas−1} for a, g ∈ L and that dimST = dimS+dimT−1 ≤ dimL−1.Then there is a basis of T of the form {g′, g′a, · · · , g′at−1} for some g′ ∈ L.

Proof. Without lost of generality, we may assume that S has a basis form {1, a, · · · , as−1}and we will proceed by induction on s = dimS ≥ 2. The case s = 2 is exactly theLemma 2.1.2. Let S′ be the subspace generated by a, · · · , as−1 then, ST = T + S′T

dimST = dim(T + S′T ) = dimT + dimS′T − dim(T ∩ S′T )

= dimS + dimT − 1 = dimS′ + dimT.

Thus, dim(T ∩ S′T ) = dimS′T − dimS′. From Theorem2.1.1 we havedimS′ + dimT − 1 ≤ dimS′T ≤ dimS′T,

dimS′T ≤ dimST = dimS + dimT − 1 = dimS′ + dimT.

Therefore, we have dimS′ + dimT − 1 ≤ dimS′T ≤ dimS′T + dimT. Setting n =dimS′ + dimT − 1 then, we have n ≤ dimS′T ≤ n + 1. Since, dimS′T ∈ N we havetwo possibilities either dimS′T = dimS′+dimT −1 or dimS′T = dimS′+dimT Thatmeans either dim(T ∩ S′T ) = dimS′ + dimT − 1 − dimS′ = dimT − 1 or dim(T ∩S′T ) = dimS′ + dimT − dimS′ = dimT. We claim that dim(T ∩ S′T ) = dimT − 1.Indeed, if dim(T ∩ S′T ) = dimT then, dimS′T = dimST. Hence, S′T = ST settingW = T + aT + · · · + as−2T we have W ⊂ ST = S′T = aW. As, dimW = dimaWit follows that, W = aW which is impossible. Given 0 < dimW < dimL and F (a)is either infinite-dimensional or equal to L. So, dimS′T = dimS′ + dimT − 1. anddimS′ = s− 1. Therefore, by induction T has a basis of the form {g′, g′a, · · · , g′at−1}for some g′ ∈ L.

2.2 Connectivity in field extension

We now transpose to the context of field extension, the basic notion of isoperimetricmethod as introduced in [4]. We will use the terminology of [5, 6] and the otheradaptation to the linear case. We know that in any field extension L/F any non-zeroF-linear form λ defines the non-degenerate symmetric bilinear form

(x, y) 7→ (x | y) = λ(xy) (2.1)

6 Chapter 2. Preliminaries and special case of Vosper’s theorem

We consider also the crucial property

(xy | z) = (x | yz) for all x, y, z ∈ L. (2.2)

In the rest of this part, we will consider an arbitrary extension L/F together witha non-zero finite dimensional F-subspace S of L. When L is itself of finite dimen-sion, we will consider it endowed with a fixed non-degenerate bilinear form (.|.)satisfying (2.1). For a subspace X ⊂ L we denote by

•X⊥ := {y ∈ L : (x|y) = 0 ∀ x ∈ X}.

• For every subspace X of L with non-zero, finite dimension we denote by

∂SX := dimXS − dimX

the increment of dimension of X when multiplied by S.

The next proposition gives us the submodularity relation.

Proposition 2.2.1 (Submodularity relation).

∂S(X + Y ) + ∂S(X ∩ Y ) ≤ ∂SX + ∂SY.

Proof. By definition we have,

∂S(X + Y ) = dim(X + Y )S − dim(X + Y )

and∂S(X ∩ Y ) = dim(X ∩ Y )S − dim(X ∩ Y ).

Using the fact that we have (X+Y )S ⊆ XS+Y S and (X ∩Y )S ⊆ XS ∩Y S we get,

∂S(X + Y ) + ∂S(X ∩ Y ) = dim(X + Y )S − dim(X + Y ) + dim(X ∩ Y )S − dim(X ∩ Y )

≤ dim(XS + Y S)− dimX − dim(XS ∩ Y S)− dimX − dimY−dimX − dimY − dim(XS ∩ Y S)

= dimXS + dimY S − dimX − dimY = ∂SX + ∂SY

Now, we are going to introduce some definitions useful for the next part of thisproject. LetXk be the set of subspacesX ofL such that k ≤ dimX <∞ and dimXS+k ≤ dimL.

Definition 2.2.2. If the set Xk is non-empty, we define thek − th connectivity of S by

Kk(S) = minX∈Xk

∂SX.

If the set Xk, is empty we set Kk(S) = −∞. When Kk(S) 6= −∞, we define a k− fragmentof S to be a subspace M of Xk with ∂SM = Kk(S). A k−fragment with minimal dimensionis called a k−atom.

Now, we are going to state an important lemma.

Lemma 2.2.3. Let the extension L/F be finite dimensional. If X is the k−fragment of S,then X∗ = (XS)⊥ is also a k− fragment of S.

2.2. Connectivity in field extension 7

Proof. We will star by remark that (X∗S)⊥ ⊇ X. In fact, let x ∈ X,x∗ ∈ X∗ and s ∈ S.

(x | x∗)s = (xs | x∗) = 0

because x∗ ∈ X∗ = (XS)⊥. Hence, dimX ≤ dim(X∗S)⊥. Knowing that, dimX∗S +dim(X∗S)⊥ = dimL then,

dimX∗S = dimL− dim(X∗S)⊥

≤ dimL− dimX. (2.3)

It follows that,

∂SX∗ = dimX∗S − dimX∗

≤ dimL− dimX − dimX∗

≤ dimL− dimX − (dimL− dimXS)

= dimXS − dimX= ∂SX (2.4)

Finally, as X satisfies dimXS ≤ dimL− k, we have

dimX∗ = dim(XS)⊥ = dimL− dimXS≥ dimL− dimL+ k

≥ k.

From (2.3), we have dimX∗S + k ≤ dimL. Joining(2.4) we can conclude that X∗ is ak−fragment of S.

Corollary 2.2.4. Let the extension L/F be finite dimensional. If A is a k−atom of S thendimL ≥ 2dimA+ Kk(S).

Proof. Set A∗ = (AS)⊥. Since dimAS = dimA + ∂SA = dimA + Kk(S) by definitionof ∂SA and Kk(S), We have dimL = dimAS+ dimA∗ = dimA+ dimA∗+Kk(S). A isa k−atom then, k−fragment. Therefore, A∗ is a k−fragment by Lemma 2.2.3. By thedefinition of k−atom we have dimA ≤ dimA∗. Hence, dimL ≥ 2dimA+ Kk(S).

The cornerstone of the method that we are going to use is the following property.

Theorem 2.2.5. Let A, B be two distinct k − atoms of S. Then dim(A ∩B) ≤ k − 1.

Proof. By the submodularity relation one has

∂S(A+B) + ∂S(A ∩B) ≤ ∂SA+ ∂SB = 2Kk(S).

By contradiction, let us suppose that dim(A ∩ B) > k − 1 i.e dim(A ∩ B) ≥ k. Bydefinition of k−atom, we have ∂S(A ∩B) > Kk(S). It follows that, ∂S(A+B) < KS .Hence,

dim(A+B)S = dim(A+B) + ∂S(A+B)

< dimA+ dimB − dim(A ∩B) + Kk(S)

< 2dimA− dim(A ∩B) + Kk(S) (2.5)= 2dimA− k + Kk(S). (2.6)

The inequality (2.5) use the fact that two k−atom have the same dimension. From(2.6) we have dim(A+B)S + k ≤ 2dimA+ Kk(S). Therefore, if dimL is infinite it is


clear that 2dimA+Kk(S) ≤ dimL. If not, from Corollary 2.2.4 2dimA+Kk(S) ≤ dimL.By definition of Kk(S), we have the contradiction because ∂S(A+B) < Kk(S).

Remark 2.2.6. If A is a k−atom of S, from the definition of ∂S and atoms it follows that foreach α ∈ L \ {0}, αA is also a k−atom of S. Therefore, there is a k−atom of S containing 1and when k = 1, by intersection theorem k = 1 the atom containing 1 is unique.

Now, we are going to state an analogue of theorem due to Mann [ch. 1, 9]. it isnot as powerful as Kneser’s theorem (see [2]). But, it is already enough to recoverthe linear Cauchy-Davenport Theorem 2.1.1.

Theorem 2.2.7. LetA be the 1−atom of S containing 1. ThenA is a subfield of L.Moreover,if

dimST < dimS + dimT − 1 < dimL, (2.7)

For some subspace T, then A is a subfield of L properly containing F.

Proof. Since A contains 1 for every a ∈ A we have a ∈ aA ∩ A. By intersectiontheorem, dim(aA ∩ A) ≤ k − 1 = 0. Hence, aA = A then, there is b ∈ A such thatab = 1. So, a−1 = b ∈ A and then, A is a subfield of L. Moreover, if dimST <dimS + dimT − 1 for some subspace T then,

0 ≤ ∂S(A) = dimAS − dimA ≤ ∂ST = dimST − dimT < dimS − 1.

If dimA = 1 then, dimAS < dimS contradiction. So, we cannot have dimA = 1.Therefore, we have the result.

Remark 2.2.8. One of the consequence of Theorem 2.2.7 is a linear Cauchy-Davenport in-equality of Theorem 2.1.1

dimST ≥ dimS + dimT − 1,

when L has no proper finite dimensional subfields containing F.

2.3 The 2 atome of Vosper space

In this subsection, we consider F to be a field and L be an extension of prime degreep of F. S, T are the subspaces of L such that 2 ≤ dimS, dimT and dimST ≤ p − 2.We are going to investigate the properties of 2−atoms of subspaces S, our goal is toshow that, under the following condition dimST = dimS + dimT − 1, the 2−atomsmust be of dimension 2.We will start by a remark, follows by two important lemmasand from these lemmas we will deduce a special case of Vosper’s theorem.

Remark 2.3.1. Let F be a finite field and let L be an extension of prime degree p of F. LetS, T be a subspaces of L such that 2 ≤ dimS, dimT and dimST ≤ p− 2. We suppose alsothat dimST = dimS + dimT − 1 then, 2−atoms of S must exists. In fact, dimST =dimS + dimT − 1 implies that ∂ST = dimST − dimT = dimS − 1.

Let L/F be an extension such that F is algebraically closed in L. If F is a finitefield and if L is of finite dimension over F, this means that L/F has a prime degree.Let S, T be a subspaces of L such that 2 ≤ dimS, dimT and dimST ≤ p − 2. Wesuppose also that dimST = dimS + dimT − 1,. In this section, the results hold inthe more general case. Let S be an F−vector space of finite dimension in L such

2.3. The 2 atome of Vosper space 9

that K2(S) = dimS − 1. Let A be a 2−atom of S, we are interested in the sequenceof subspaces Ai for i ≥ 1, where Ai is defined inductively by Ai = Ai−1A. In thissection, we shall show that A is also a 2− atom of A, that Ai, i ≥ 1 is a 2−fragmentof A as long as dimAiS+ 2 ≤ dimL, and that in the case when dimL is finite dimL ≡2mod(n− 1), where n = dimA. Let state the following lemmas useful in the proof ofthe main result.

Lemma 2.3.2. If A is a 2−atom of S then A is a 2−atoms of A.

Proof. If dimA = 2 then, dimAA + 2 ≤ dimL. Suppose that dimA = n ≥ 3. Notethat, S is a witness that K2(A) = dimA − 1. Let B a 2−atom of A. Without loss ofgenerality, one can suppose that 1 ∈ A ∩B. Let b ∈ B\F and a ∈ A\F.

Claim 2.3.3. dimA = dimB and AB = A+ bA = B + aB.

Proof. dim(A + bA) = dimA + dimbA − dim(A ∩ bB) bB is also a 2−atom of A. Byintersection theorem dim(A ∩ bB) ≤ 2− 1 = 1. Hence,

2dimA− 1 ≤ dim(A+ bA) ≤ dimAB = dimA+ dimB − 1.

Since, A + bA ⊆ AB we have dim(A + bA) ≤ dimAB therefore dimB ≥ dimA. Bythe same way, we have 2dimB − 1 ≤ dim(B + aB) ≤ dimAB = dimA+ dimB − 1.Then, dimB ≤ dimA. So, dimA = dimB. Moreover AB = A+ bA = B+ aB because1 ∈ A ∩B.

Claim 2.3.4. dimA2B < dimL.

Proof. We have, A2B = A(AB) = A(aB +B) ≤ aAB +AB. Hence,

dimA2B ≤ 2dimAB − dim(aAB ∩AB)

using the fact that aB ⊆ aAB ∩AB we have:

dimA2B ≤ 2(dimA+ dimB − 1)− dimB= 2dimA+ dimB − 2 (2.8)= 3n− 2. (2.9)

From Corollary 2.2.4 we have: dimL ≥ 2dimA+K2(A) = 2n+n−1 = 3n−1.Hence,dimL > dimA2B.

The Claim 2.3.4 allows us to apply the linear Cauchy-Davenport Theorem 2.1.1to the space A2 and B. And together with (2.8) we obtain

dimA2 + dimB − 1 ≤ dimA2B ≤ 2dimA+ dimB − 2.

Which implies dimA2 ≤ 2dimA− 1. This shows that A is its own 2−atom.

Lemma 2.3.5. Let A be a 2−atom of S and t ≥ 2 an integer. We have

dimAt = min{dimAt−1 + dimA− 1, dimL}.

Proof. Let a ∈ A \ F. By Lemma2.3.2 A is a 2−atom of A and

dimA2 = dim(A+ aA) = 2dimA− 1,


which establishes the result for t = 2. In particular, A2 = A+ aA, so that we have

At = At−2(A+ aA) = At−1 + aAt−1.

Notice that At−1 ∩ aAt−1 contains aAt−2, therefore.

dimAt = dim(At−1 + aAt−1)

= 2dimAt−1 − dim(At−1 ∩ aAt−1)≤ 2dimAt−1 − dimAt−2.

By induction hypothesis, we have dimAt−1 = dimAt−2 + dimA − 1 this meansdimAt−1 − dimAt−2 = dimA − 1. Therefore, dimAt ≤ dimAt−1 + dimA − 1. SinceAt 6= L, the linear Cauchy-Davenport inequality gives us the following inequality.dimAt−1 + dimA− 1 ≤ dimAt which yields dimAt = dimAt−1 + dimA− 1.

Lemma 2.3.6. Let A be a 2−atom of S with dimA = n > 2. Then, if dimL is finite wehave dimL ≡ 2mod(n− 1).

Proof. Let t be the largest positive integer such that dimAt < dimL. For sure, thelargest exists because the dimension of L is finite. Since, A is also a 2−atom of A wehave dimA2 ≤ dimL − 2, so that t ≥ 2. Let X = At−1 and let X∗ = (At)⊥. As, wehave supposed dimAt < dimL, we have dimX∗ > 0. From Lemma 2.3.5 it followsthat, dimAt = dimAt−1 + dimA − 1. By our assumption, we have dimAt < dimL.Now, if dimAt + 1 = dimL then, dimX∗ = 1. If not, dimAt + 2 ≤ dimL. Therefore,At−1 is a 2−fragment of A, for t ≥ 2 we have for free that dimAt−1 ≥ 2. From theLemma 2.3.5 we have either dimX∗ = 1 or X = At−1 is a 2−fragment of A.

Claim 2.3.7. We claim that is not possible to have X = At−1 to be 2−fragment of A.

By contradiction, let suppose that X it is. Then, the maximality of t such thatdimAt < dimL and Lemma 2.3.5 imply that dimX∗ ≤ dimA − 1. By Lemma 2.2.3X∗ = (At−1A) is a 2−fragment of A. As, At−1 is a 2−fragment of A, we havedimX∗ ≤ dimA which is a contradiction because, A is a 2−atom of A by Lemma2.3.2. So, dimX∗ = 1 and therefore dimAt = dimL − 1. From Lemma 2.3.5, one candeduce that dimAt = dimA+ (t− 1)(dimA− 1), and

dimL = dimA+ (t− 1)(dimA− 1) + 1

= dimA+ tdimA− t− dimA+ 2

= t(dimA− 1) + 2.

Then, dimL ≡ 2mod(dimA− 1).

The Lemma2.3.6 above has an important consequence. We are going to state inthe next step and from this consequence will derive a particular Vosper’s theorem.

Consequence 2.3.8. If dimL − 2 = m − 2 is prime, then m − 2 = t(dimA − 1) implieseither t = m− 2 and dimA− 1 = 1 or t = 1 and dimA− 1 = m− 2. The case t = 1 anddimA− 1 = m− 2 is impossible because t ≥ 2. So, dimA = 2.

We are going to state and give the prove of Vosper’s theorem is some particularcase. Indeed, we have all the ingredients to deduce the result.

Theorem 2.3.9. Let F \ L be a finite extension such that F is algebraically closed in L.Suppose that [L : F ]− 2 is prime. Let S, T be a subspaces of L such that 2 ≤ dimS, dimT

2.3. The 2 atome of Vosper space 11

and dimST ≤ dimL − 2. If dimST = dimS + dimT − 1 then, there are bases of S andof T respectively of the form

{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1

}for some

g, g′, a ∈ L.

Proof. Let A be a 2−atom of S, from the Consequence 2.3.8 the dimension of A is 2.And by the Lemma2.1.2 and Lemma2.1.3 the result holds.

The aim of this chapter, was to present some basic materials that we are going inthe rest of this project. The second aspect was to establish the Vosper’s theorem inthe case of an extension L of F,with F algebraically closed in L and [L : F ]−2 primeinteger. In the next chapter, we are going to investigate the case where F finite field,and L a finite extension of F. Using some materials define in this chapter, and thesome new tools that we will introduce.

12

Chapter 3

Connection between Quadraticform and Vosper’s theorem andApplication to finite field

This chapter, has three principal aspects. The first one, it is to explain why it isimportant to think about finite field and algebraically field. The second point of thischapter, is to present the connection between quadratic form and Vosper’s theorem.The last one, it is to redefine our problem in the space of quadratic forms. Since ourgoal is to show the Vosper’s theorem, we will show that, in the case of finite fieldthe result holds. The main idea come from [1]. We will take F to be a finite fieldand L is a finite extension of F. The key point will be to show that the dimension of2−atoms is equal to 2. To deal with this problem, we will use the new definition ofour problem into a space of quadratic forms.

3.1 Motivation, Sidon spaces and Quadratic forms

We will start by giving a proposition, which is in fact a motivation to study theVosper’s theorem in the case of finite field and others fields satisfying some condi-tions, which will be state in the following proposition. Before giving the statementof the proposition, let give the following definition.

Definition 3.1.1. We denote by Qn the vector space of homogeneous polynomials of degree2 in the variables x1, · · · , xn with coefficient in the field F. A typical element of Qn will bedenoted by

Q =∑

1≤i≤j≤naijxixj . (3.1)

The F−vector space Qn is of dimension n(n + 1)/2 and we can identify it withthe space of quadratic forms over Fn. Similarly, let Ln denote the space of linearforms over Fn, identified with the space of homogeneous polynomials of degree 1in x1, · · · , xn.

Proposition 3.1.2. Let F be a field, if there is a quadratic form in three variables withoutnon trivial zero (anisotropic) then, there exist an extension L of F and A ⊂ L satisfyingdimA2 = 2dimA− 1 such that, A does not have basis in geometric progression.

Proof. Let Q(x, y, z) be a quadratic polynomial in three variables without non trivialzero. Let P (x, y) = Q(x, y, 1). It is clear that P is irreducible over F otherwise Qwould have a non trivial zero. Hence, we can take L = F (x)[y]/〈P (x, y)〉. Let A =span{1, x, y}, the span(A2) = {1, x, y, x2, xy, y2} from P (x, y) = 0 we can express y2

3.1. Motivation, Sidon spaces and Quadratic forms 13

in terms of 1, xy and x2. It follows that, dimA2 ≤ 5. By Cauchy-Davenport theorem,we have dimA2 ≥ 2dimA − 1 = 5 then, it follows that dimA2 = 5 = 2dimA − 1. So,dimA2 = dimA − 1. We claim that A does not have basis in geometric progression.By contradiction, let assume A has basis in geometric progression equivalent to saythat A = span〈u, ug, ug2〉 with some u, g ∈ L. Then, there exist αi, βi, and γi withi = 1, 2, 3 such that l1 := α1 + β1x + γ1y = u, l2 := α2 + β2x + γ2y = ug, l3 :=α3 + β3x+ γ3y = ug2. In L we have (l2)

2 = l1l2. Hence, (l2)2 = l1l2 + T (x, y)P (x, y).

Since the degree of the left hand side is 2 then, λ = T (x, y) should be a non zeroconstant. Switching in the projective coordinate we have,

λQ(x, yz) = (α2z + β2x+ γ2y)2 − (α1z + β1x+ γ1y) (α3z + β3x+ γ3y) . Since,the number of variables is equal to three, the equations α2z + β2x + γ2y = 0 andα1z + β1x + γ1y = 0 have a non trivial common zero and then, Q(x, y, z) also. Thatcontradict the fact that, Q does not have a non trivial zero.

This proposition allows us to say, the Vosper’s theorem it is not always true inany field and then, we have a necessary condition to have it in some field. In the fol-lowing subsection, we are going to give the link between Sidon space and Quadraticform.

3.1.3 Sidon spaces and quadratic form

The Intersection Theorem gives a key property of 2−atoms that in extension L/Fwithout proper finite sub-extensions of F, translates into:

For all x ∈ L/F, dim(A ∩ xA) ≤ 1 (3.2)

We note that (3.2) implies that for all x, y, z, t ∈ A \ {0} xy = zt =⇒ {Fx, Fy} ={Fz, F t}. This is because since x−1A∩ z−1A contains F, we deduce that either x andz are F−proportional from which the conclusion follows, or that x−1A ∩ z−1A = Fby (3.2)from which it follows that x−1A and z−1A each generate the constant field F.

Definition 3.1.4. Let L/F be an extension. The subspace A of L satisfying (3.2) is calledthe Sidon space.

The Definition 3.1.4 is just an analogy with classical Sidon sets. Let state thefollowing remarks.

Remark 3.1.5. When A has a basis in geometric progression then, there exists an x ∈ Lsuch that dim(A∩xA) = dimA−1. Therefore, Sidon space can be thought of as spaces thatare "furthest away" from a space with a basis in geometric progression.

Remark 3.1.6. For a Sidon space we have

dimA2 ≥ dim(A+ aA) ≥ 2dimA− 1. (3.3)

For any a ∈ A \ F.Consequence 3.1.7. According to Lemma2.3.2, if A is a 2−atom of some set S satisfyingdimSA = dimS + dimA − 1 without proper finite subextension then, the inequalities in(3.3)are actually equalities.

We are going to recap what we have said by the following theorem.

Theorem 3.1.8. Let L/F be an extension and suppose F algebraic closed in L. Let S be asubspace of L with finite dimension greater or equals than two. And let A be a 2−atom ofS then A, is a Sidon space. Furthermore if dimSA = dimS + dimA − 1 then dimA2 =2dimA− 1.

14Chapter 3. Connection between Quadratic form and Vosper’s theorem and

Application to finite field

Now, classical Sidon sets S of integers (Sidon sets in abelian groups) have the

property that |S + S| =(| S | +1

2

).

Which implies in particular that, | S + S |= 2 | S | −1 if and only if | S |= 2.Indeed, if | S |≤ 2 we have(

| S | +12

)=

(| S | +1)!

2(| S | −1)!(3.4)

=1

2(| S | +1) | S | . (3.5)(

| S | +12

)= 2 | S | −1 if and only if (| S | +1) | S |= 4 | S | −2. Hence,(

| S | +12

)= 2 | S | −1 if and only if | S |2 −3 | S | +2 equivalent to | S |≤ 2.

If we let ourselves train by the additive analogy we may be led to expect for a mo-

ment that for any Sidon space A it is true that dimA2 =

(dimA+ 1

2

). This would

immediately implies that, the only Sidon spaces A such that dimA2 = 2dimA− 1 areof dimension d = 2. Actually, is not true in general we are going to give a counterexample.

Counter-Example 3.1.9. Let F the finite field of size 2, for L the field of size 219, and letA be the F−vector with basis (1, α, α7, α12 + α3 + 1) where α is a root of the irreduciblepolynomial X19 +X14 +X10 +X7 +X2 +X + 1.

Then the dimension of A2 = 9 <

(dimA+ 1

2

).

Key point to achieve the main result of this thesis is only to show that the onlySidon spaces satisfy (3.3) are the dimension of dimension d ≤ 2. If F is allowed tobe any field, again this is not true.

Theorem 3.1.10. If L/F is an extension with F algebraic closed in L, such that for a Sidonsubspace A of L of finite dimension dim ≥ 3, we have, dimA2 > 2dimA − 1. Then, theVosper’s theorem holds.

Remark 3.1.11. The Theorem 3.1.8 together with the hypothesis of Theorem 3.1.10 willshow that the dimension of A is 2.

Let build the connection between Theorem 3.1.10 and quadratic form. Actually,we shall transform the problem into a problem in the space of quadratic forms. Letus now, introduce a notion of weight of a quadratic form.

Definition 3.1.12. For a non-zero quadratic form Q ∈ Qn, let its weight equals to thesmallest integer k such that Q can be expressed as a sum of k products of linear forms inx = (x1, · · · , xn).

Wt(Q) := min{k : Q = l1(x)l′1(x)+ l1(x)l′1(x)+ · · ·+ lkl′k(x), li, l′j ∈ Ln, 1 ≤ i, j ≤ k}.

If C is a set of quadratic forms, we will call the minimum weight of C the smallest weight ofthe difference between two distinct quadratic forms of C :

Wt(Q) := min{Wt(Q−Q′) : (Q,Q′) ∈ C × C, Q 6= Q′}.

We note that, if C is a subspace of Qn then, Wt(C) = min{Wt(Q) : Q ∈ C, Q 6= 0)}.

3.2. Codes in the space of quadratic forms over a field 15

Now, let A be a Sidon space of dimension n in some extension of F. And let(a1, · · · , an) be a basis of A. Consider the homomorphism Φ of spaces Ln → A de-fined by the mapping

x1 7→ a1, · · · , xn 7→ an.

This homomorphism extends in a natural way to a homomorphism of F−vectorspaces through the relation xixj 7→ aiaj . Note that for any l, l′ ∈ Ln, the map Φsatisfies Φ(ll′) = Φ(l)Φ(l′). Consider the subspace C of Qn, C = kerΦ.

Proposition 3.1.13. A is a Sidon space if and only if for any Q ∈ C, Q 6= 0, we haveWt(Q) ≥ 3.

Proof. We will start by remark that Wt(Q) > 1. Because the element of A2 live in afield where the products of non-zero elements are non-zero. Suppose A is a Sidonspace. IfWt(Q) = 2,meaningQ = l1l

′1 + l2l

′2, then setting x = Φ(l1), y = Φ(l′1), z =

Φ(l2) and t = Φ(l′2). Hence, we have Φ(Q) = 0 is equivalent to xy + zt = 0 inA2. A being a Sidon space, according to the remark following (3.2), that either x isan F−multiple of z and y an F−multiple of t or x is an F−multiple of t and y anF−multiple of z. Since the map Φ is one-to-one from Ln to A we deduce from thisthat Q is F−proportional to l1l′1. Which contradicts the fact Wt(Q) = 2. Conversely,suppose that for any Q ∈ C Q 6= 0 Wt(Q) ≥ 3 then, A is a Sidon space.

Now, we are going to give an example of Sidon space in a finite extension.

Example 3.1.14. Let L = Q(a), where a is a root of the irreducible polynomial p(x) =2 + 2x + 2x2 + 2x5 + x8 = 1 + x2 + (1 + x2 + x4)2 it Galois group of this polynomialis the full symmetric group S8. Which implies that the extension L/Q has no intermediateextension. Let A be a the subspace with basis (1, a, b) where b = 1 + a + a2, becauseb2 = (1 + a + a4) = −

(1 + a2

)we have dimA2 = 5. We know also that for the quadratic

forms in three variables the map Φ maps x 7→ a, y 7→ b, and z 7→ 1. The dimension of KerΦis equal to 1 and is generated by a quadratic form x2 + y2 + z2 which is a weight 3. ByProposition3.1.13 is a Sidon space.

It is important to know that dimA2 = 2dimA−1 is equivalent to say that dimkerΦ =n(n + 1) − (2n − 1) = (n − 1)(n − 2)/2. This it is the same thing to say, the hypoth-esis of Theorem3.1.10 should hold if we can show that there is no subspace of Qn ofquadratic form of minimum weight 3 with dimension (n− 1)(n− 2)/2.

Remark 3.1.15. Our new problem consist to show the non existence of subspace of Qn ofquadratic form of minimum weight 3 with dimension (n− 1)(n− 2)/2.

3.2 Codes in the space of quadratic forms over a field

3.2.1 Action ofMn(F ) on quadratic forms

Let x = (x1, · · · , xn), and Mn(F ) be the ring of n × n matrices with entries in F.Given U = (uij) ∈Mn(F ) and any quadratic form Q(x) = Q(x1, · · · , xn), we defineanother quadratic form

QU (x) = Q(U>x) = Q (u11x1 + · · ·+mn1, · · · , u1nx1 + · · ·+mnn) .

This is an action in the sense that

(i) QIn(x) = Q(x).



(ii) For all U1, U2 ∈Mn(F ), we have QU1(QU2) = QU1U2 .

Restricting to GLn(F ), the group of invertible matrices, we have a group action.

Definition 3.2.2. We say that two quadratic form Q and Q′ are equivalent, if there existM ∈ GLn(F ) such that QM = Q′. This is evidently an equivalence relation in which theequivalent classes are precisely the GLn(F )−orbits. More generally, any subgroup G ⊂GLn(F ) certainly acts well, and we can define two quadratic forms to G-equivalent if theylie in the same G−orbit.

Remark 3.2.3. If we consider G = F∗q×GLn(F ), where F∗q acts by scalar multiplication onquadratic forms and the linear groupGLn(F ) For g = (a, U) ∈ G = F∗q×GL(n, q), a rightgroup action is thus define on elements Q by Qg = aQU . The orbits of Qn under this actionare straightforwardly obtained from the well known description of the Qn under GLn(F ).

Definition 3.2.4. The rank of Q is the codimension of its radicalRadQ, the linear spaceof x ∈ Fn, such that Q(x) = 0 and BQ(x, y) = 0 for all y ∈ Fn, where BQ(x, y) =Q(x+ y)−Q(x)−Q(y) is the symmetric bilinear form associated to Q. A quadratic formQ ∈ Qn is said to be non degenerate if rk(Q) = n.

Definition 3.2.5. We denote by

Ax1,x2 : = {Q = x1(x)l′1(x) + x2(x)l′2(x) : l′i ∈ Ln} ⊂ Qn (3.6)

the subspace ofQn.

Remark 3.2.6. If U ∈ Gln(F ) such that lU1 (x) = l1(Ux) = x1 and lU2 (x) = l2(Ux) = x2then, Aul1,l2 = Ax1,x2 . Al1,l2 is a subspace of dimension 2n− 1.

Proposition 3.2.7. Let n ≥ 3, and let C ⊂ Qn, be a linear code with Wt(C) ≥ 3. Then,

dimC ≤ (n− 1)(n− 2)

2(3.7)

Proof. For every Q ∈ Al1,l2 , we have Wt(Q) ≤ 2 then C ∩ Al1,l2 = {0}. Hence,dimC ≤ dimQn − dimAl1,l2 = (n−1)(n−2)

2 .

To deal with the problem, we should actually show that is not possible to havedimC = (n−1)(n−2)

2 . By contradiction, we will assume that dimC = (n−1)(n−2)2 equiv-

alent to say C ⊕ Al1,l2 = Qn this means also that C⊥ ∩ A⊥l1,l2 = {0}. Let us recall thedefinition of paring.

Definition 3.2.8. A pairing (., .) : Qn × Sn → C∗ is an application which respect to theaction of with respect to Q and B meaning that

(i) (Q+Q′, B) = (Q,B)(Q′, B) for all Q,Q′ ∈ Qn and B ∈ Sn.

(ii) (Q,B +B′) = (Q,B)(Q,B′) for all Q ∈ Qn and B,B′ ∈ Sn.

The pairing is said to be non degenerate if (Q,B) = 1 for all B ∈ Sn implies that Q = 0,and similar for B ∈ Sn.

Let A := Ax1,x2. We note that Ag runs over the set {Al1,l2 : (l1, l2) independent}.Going back to definition of A and the Definition 3.2.8 we see that, A⊥ = {B′ ∈ Sn :B′1,j = B′2,j = 0 for all 1 ≤ j ≤ n}. So, the orbit of B intersects A⊥ if and only ifrk(B) ≤ n− 2. We can resume what we have done by the following proposition.

3.3. Vosper’s theorem for finite field 17

Proposition 3.2.9. (i) C is a linear code such that Wt(C) ≥ 3 if and only if rk(C⊥) ≥n− 1.

(ii) dimC = dimQn − (2n− 1) if and only if dimC⊥ = 2n− 1.

Remark 3.2.10. The new problem consist to show that if B ∈ Sn is a linear code, of ranggreater or equals to n− 1 and dimB = 2n− 1 then n = 2.

In the next section we are going to solve this question in the case of finite field.

3.3 Vosper’s theorem for finite field

In this section, we take F = Fq where q is a power of a prime integer. We will start bydescribing the notion of G−equivalence in the case where G = F∗q × GLn(F ). Overfinite fields, the G−orbit of quadratic forms not characterised solely by its rank. Itturns out that, the set of quadratic forms of given rank r makes up one orbit when ris odd, but splits into two orbits when r is even:

Proposition 3.3.1. If Q is a quadratic form over Fq in n variables of rank r = rk(Q) > 0,then one of the following holds:

(1) r is odd and Q v∑(r−1)/2

i=1 x2i−1x2i + x2r .

(2) r is even and Q v∑r/2

i=1 x2i−1x2i.

(3) r is even and Q v∑( r

2−1)

i=1 x2i−1x2i +Q0(xr−1, xr), where

Q0(x1, x2) :=

{x21 − bx22 if p 6= 2

x21 + x1x2 + bx22 if p = 2(3.8)

In the above, b ∈ Fq is such that b ∈ Fq \ F2q if p 6= 2, and b ∈ Fq \ σ(Fq) if p = 2,

where σ(Fq) = {λ2 + λ, λ ∈ Fq)}.

Remark 3.3.2. A quadratic form over F is either equivalent to hyperbolic form (this is case(2)), or the direct sum of a hyperbolic form and an anisotropic form, i.e a quadratic formwithout non trivial zeroes, in one (case(1)) or two variables (case 3).

Notation 3.3.3. If Q is a quadratic form of rank r, and if e ∈ {0, 1, 2} denotes the rank ofits anisotropic component, we will say that Q has type (r, e) and we will write Q v (r, e).

Now, we are going to make the connection between the weight and the type ofquadratic form. Since, the weight of a quadratic form is left unchanged under theaction of G, it should be expressible in terms of its type.

Lemma 3.3.4. If Q v (r, e), Wt(Q) = r+e2 .

Proof. If Q is hyperbolic of rank 2h, h ∈ N meaning that Q v∑r/2

i=1 x2i−1x2i. So, theWt(Q) = h = r/2. Whence e = 1, because Wt(x2r) = 1 we have, Wt(Q) = r−1

2 +1 = r+1

2 . Whence e = 2, it is immediate that, with the notation of Proposition3.3.1,Wt(Q0) = 2, Wt(Q) = ( r2 − 1) + 2 = r+2

2 .

The space Sn of symmetric bilinear form over Fq is also equipped with the natu-ral action of G, given by Bg = aU>BU where, we identify symmetric bilinear formsand symmetric matrices. We notice that, the Fq−vector space Qn and Sn have the



same dimension n(n + 1)/2. When q is odd, the correspondence Q 7→ BQ definesan isomorphism of G−spaces, because Q can be recovered from BQ thanks to theformula BQ(x, x) = 2Q(x). But when q is even, BQ is alternating and the correspon-dence Q 7→ BQ is not one to one. However, for all q there exist a non degeneratepairing between Qn and Sn that behaves well with respect to the action of with re-spect to Q and B.

Lemma 3.3.5. Let α : (Fq,+)→ (C∗,×) be a fixed non trivial character. Let, for Q ∈ Qn,Q =

∑1≤i≤j≤n ai,jbi,jB ∈ Sn,

(Q,B) = α

∑1≤i≤j≤n

ai,jbi,j

.

Then, this expression defines a non degenerate pairing betweenQn and Sn. Moreover, for allQ ∈ Qn, B ∈ Sn, g ∈ G, we have

(Qg, B) = (Q,Bgt) (3.9)

where, if g = (a, u) ∈ G, we denote gt = (a, ut).

The pairing introduced above allows for a convenient description of the multi-plicative characters of the additive group (Qn,+), and of the way the group acts onthem. Indeed, the character of Qn are in one correspondence with Sn and for everyB ∈ Sn the associated character is given by:

χB(Q) = (Q,B)

whereB ∈ Sn. Furthermore, if we define the action ofG on characters by (g.χ)(Q) :=χ(Qg), the Lemma 3.3.5 translates into:

g.χB = χBgt .

In the next proposition, we recall the description of the orbits of Sn under the action

ofG.A matrix with a diagonal block structure(A 00 B

)is denoted below byA

⊕B.

Proposition 3.3.6. If B ∈ Sn is symmetric bilinear form over Fq in n variables of rankrk(B) = r > 0, then one of the following holds:

(1) r is odd and B v⊕ r−1

2i=1

(0 11 0

)⊕(1).

(2) r is even and B v⊕r/2

i=1

(0 11 0

).

(3) r is even and B v⊕ r

2−1

i=1

(0 11 0

)⊕B0. Where B0 =

(0 11 b

)if p = 2, and B0 =(

1 00 −b

)b ∈ Fq \ F2

q if p 6= 2.

Similarly to quadratic forms, we say that a symmetric bilinear form B is of thetype (r, e), and write B v (r, e), when r is the rank of B, and e takes the valuee = 1, 0, 2 when B is in the case (1), (2), (3), respectively. If in addition we define the


type of Q = 0, and of B = 0, to be 0, we obtain a convenient parametrization of theorbits of Qn and of Sn under the action of G by the same set Tn :

Tn := {0} ∪ {(r, e) : 1 ≤ r ≤ n, e ∈ {0, 1, 2}, r ≡ e mod 2, r ≤ e}.

When dealing with types, orbits or stabilizers, we will use the same set of notationregardless of whether we are speaking of quadratic forms or of symmetric bilinearforms. However, to avoid with confusion, we will preferentially reserve the letter tfor types of elements of Qn and the letter s for the types of elements of Sn.

Let Xt =| C ∩ Ot |=| {Q ∈ C : Q v t} | where C ⊂ Qn denote a subspace of Qn,such that Wt(C) ≥ 3. In the following subsection, we will find the equation satisfiedby {Xt, t ∈ Tn} and try to simplify the system in a new system with few equations.

3.3.7 The equations satisfied by {Xt, t ∈ Tn}

In this subsection, we will be concerned by the unknowns Xt, for t ∈ Tn. And there

satisfy a few trivial equations:

X0 = 1∑

t∈Tn Xt = |C|Xt = 0 for t = (1, 1), (2, 0), (2, 2), (3, 1), (4, 0).

The

last set of conditions follows from the fact that, we are working with the quadraticforms of Wt(Q) ≥ 3. And the Lemma3.3.4 tell us that the weight of Q is 1 or 2 whent = (1, 1), (2, 0), (2, 2), (3, 1), (4, 0), therefore Xt = 0. Now, we are going to showthat if the code C satisfies equality in (3.7) then, some additional equations hold for{Xt}. First we need to introduce some notations, before the notations let us state thefollowing remark.

Remark 3.3.8. For s ∈ Tn, the expression∑B∈Os

χB(Q) =∑B∈Os

(Q,B)

only depends on the type of Q. Indeed, this property follows immediately from (3.9).

This remark justifies the following notation

χs(t) :=∑B∈Os

(Q,B) (Q ∼ t). (3.10)

Proposition 3.3.9. LetC ⊂ Qn, a linear code, such thatWt(C) ≥ 3 and |C| = q(n−1)(n−2)

2 .For all s = (r, e) such that 1 ≤ r ≤ n− 2, we have∑

t∈Tn

χs(t)Xt = 0.

Proof. We consider the function on Qn defined by:

F (Q) :=∑g∈G

1Ag(Q)



where A := Ax1,x2. We note that Ag runs over the set {Al1,l2 : (l1, l2) independent}.We will compute in two different ways the expression:∑

: =∑

Q,Q′∈C2

F (Q−Q′). (3.11)

The first way is straightforward, if Q 6= Q′, the weight of Q − Q′ is at least equal to3, consequently Q−Q′ cannot belong to Ag. So,∑

: =∑Q∈C

F (0) = |G||C|. (3.12)

The second method uses the expansion ofF over the characters ofQn, in other wordsits Fourier expansion. We introduce the complex vector space

L(Qn) := {f : Qn → C}.

The multiplicative characters χB : (Qn,+) → C∗, for some B ∈ Sn form a basis ofL(Qn), which is orthogonal for the standard inner product

〈f1, f2〉 : =1

|Qn|∑Q∈Qn

f1(Q)f2(Q). f1, f2 ∈ L(Qn). (3.13)

So, we can write F =∑

B∈Sn fBχB. It will be essential in what follows that fB ≥ 0,and also need to know when fB > 0. We have the answer in the next lemma.

Lemma 3.3.10. With the above notation, fB ≤ 0 for all B ∈ Sn, and fB > 0 if and only ifrk(B) ≤ n− 2. Moreover, f0 = |G||A|

|Qn| .

Proof. We have

fB = 〈F, χB〉 =1

|Qn|∑Q∈Qn

F (Q)χB(Q)

=1

|Qn|∑Q∈Qn

∑g∈G

1Ag(Q)χB(Q)

=1

|Qn|∑Q∈Qn

∑g∈Ag

χB(Q).

Let A⊥ = {B ∈ Sn : (Q,B) = 1 for all Q ∈ A}. Then, from the property of thepairing, (A)⊥ = (A⊥)g

−t(where g−t denotes the transpose of the inverse of g), and

because Ag is a subgroup of Qn,

∑Q∈Ag

χB(Q) =

{0 if B /∈ (A⊥)g

−t

|Ag| = |A| if B ∈ (A⊥)g−t.

So, we find

fB =|A||Qn||{g ∈ G : Bgt ∈ A⊥}|. (3.14)

From (3.14), it is clear that fB ≥ 0. Moreover, we see that fB > 0 if and only if theorbit of B intersects A⊥. The expression for f0 follows from (3.14).


Going back to∑

in (3.11), we have:

∑=

∑B∈Sn

fB

∑(Q,Q′)∈C2

χB(Q−Q′)

(3.15)

=∑B∈Sn

fB|∑Q∈C

χB(Q)|2 ≥ f0|C|2 (3.16)

To get the last inequality, we have neglected the contributions of all the charactersexcept that of the trivial one, the non negativity of the coefficients fB is crucial inthis step. If we recap everything from (3.12), (3.16) and Lemma 3.3.10, we obtain

|G||C| =∑≥ |G||A|Qn

|C|2.

After simplification we obtain the following inequality 1 ≥ |A|Qn|C|, this equality is

nothing more that (3.7). The interesting point is when we have the equality 1 =|A|Qn|C| holds if and only if the neglected terms in (3.11) are equal to zero, leading, in

the case of a hypothetical optimal code, to the conditions∑Q∈C

χB(Q) = 0 iffB > 0, B 6= 0. (3.17)

From the Lemma3.3.10, we know that fB > 0 if and only if rk(B) ≤ n − 2. Let s =(r, e) where 1 ≤ 1 ≤ n−2, observing that the condition fB > 0 holds simultaneouslyfor all the elements of the orbits Os allows us to sum up the equation(3.17) overB ∈ Os. The expression ∑

B∈Os

χB(Q) =∑B∈Os

(Q,B)

only depends on the type of Q, since∑B∈Os

(Q,B) =∑B∈Os

(Q,Bgt

)=∑B′∈Os

(Q,B′

).

So, the Definition 3.10 is consistent and we obtain that

∑t∈Tn

χs(t)Xt =∑B∈Os

∑Q∈C

χB(Q)

= 0.

We can recapitulate what we have achieved by the following proposition.

Proposition 3.3.11. Suppose C ∈ Qn is such that 0 ∈ C, |C| = q(n−1)(n−2)

2 and Wt(C) ≥3. Then,

Xt := |C ∩ Ot| (t ∈ Tn)



satisfy the following equations:

(S) :

(i) X0 = 1

(ii) Xt = 0 for t = (1, 1), (2, 2), (3, 1), (4, 0)

(iii)∑

t∈Tn Xt = q(n−1)(n−2)

2

(iv)∑

t∈Tn χs(t)Xt = 0 for all s = (r, e), 1 ≤ r ≤ n− 2.

(3.18)

The system (S) is a linear system with |Tn| unknowns (the variableXt) and |Tn|+3 equations. So, if the system (S) does not have a solutions then we have proved thenon existence of the code. To make the resolution of the system (S) above easy, weare going the make some change of variables in the next subsection to have a smalllinear system.

3.3.12 Change of variables

Let us start by the following proposition.

Proposition 3.3.13. Suppose C ∈ Qn is such that 0 ∈ C, |C| = q(n−1)(n−2)

2 and Wt(C) ≥3. Then,

Xt := |C ∩ Ot| (t ∈ Tn).

And letYs =

1

|C|∑t∈Tn

χs(t)Xt (t ∈ Tn).

The number Ys satisfy the following equations:

(S′) :

(i′) Y0 = 1

(ii′) Ys = 0 for all s = (r, e), 1 ≤ r ≤ n− 2.

(iii′)∑

s∈Tn Ys = q2n−1

(iv′)∑

s∈Tnχs(t)|Os| Ys = 0 for t = (1, 1), (2, 0), (2, 2), (3, 1), (4, 0)

(3.19)

Proof. The new variables Yt are related to the coefficients {λB, B ∈ Sn} of character-istic function 1C of C on the basis of characters:

1C =∑B∈Sn

λnχB. (3.20)

Indeed, we have

λB = 〈1C , χB〉 =1

Qn

∑Q∈C

χB(Q) (3.21)

so ∑B∈Os

λB =1

Qn

∑Q∈C

( ∑B∈Os

(Q,−B)

)=

1

Qn

∑t∈Tn

χs(t)Xt

and thus

Ys =|Qn||C|

∑B∈Os

λB. (3.22)


Let us verify the equation of the system S′. From (3.22), Y0 = |Qn||C| λ0 and from(3.21)

λ0 = |C||Qn| so we find (i′). The equation (ii′) follows immediately form (iv). From

(3.21) and (3.20), we have (iii′) :∑sTn

Ys =|Qn||C|

∑B∈Sn

λB =|Qn||C|

1C(0) = q2n−1.

It remains to prove (iv′). To end with this, we introduce the characteristic function1Ot of quadratic forms of type t and its decomposition as a linear combination ofcharacters:

1Ot =∑B∈Sn

µt,BχB,

whereµt,B = 〈1Ot , χB〉 =

1

|Qn|∑Q∈Ot

χB(Q) =1

|Qn|∑Q∈Ot

χB(Q).

The last equality holds because for any Q ∈ Ot we have −Q ∈ Ot and χB(−Q) =(−Q,B) = (Q,B) by Definition3.2.8. Now, the expression

∑Q∈Ot

χB(Q) dependsonly on the type s of B, and

|Os|

∑Q∈Ot

χB(Q)

=∑B∈Os

∑Q∈Ot

χB(Q)

=

∑B∈Ot

∑Q∈Os

χB(Q)

=∑Q∈Ot

χs(t) = |Ot|χs(t).

So,

µt,B =|Ot|χs(t)|Qn||Os|

. (3.23)

Then,Xt = |Qn〈1Ot , 1C〉 = |Q|

∑s∈Sn

µt,BλB.

Taking account of (3.23) and (3.22), we find

Xt = |Ot|∑B∈Sn

χs(t)

|Os|λB = |Os|

∑B∈Tn

χs(t)

|Os|

( ∑B∈Os

λB

)(3.24)

=|Ot||C||Qn

∑B∈Tn

χs(t)

|Os|Ys. (3.25)

The above relation between Xt and Ys show that (ii) is equivalent to (iv′).

Now, we are going to solve the system actually, we are going to show that for n ≥3 the system (S′) does not have a solution. It is important to know that, the system(S′) it is more soft that (S) because it can re-written as a linear system in only threevariables either {Y(n−1,0),Y(n,0)

, Y(n−2)} or {Y(n−1,0), Y(n−1,2), Y(n,1)}, depending on theparity of n and six equations. In order to prove solve our problem it will be enough totake account of the equations (iv′) associated to the type t = (1, 1), (2, 0), (2, 2), and



thus compute the values of χs(t) for s = (n, e) (n − 1, e) and t = (1, 1), (2, 0), (2, 2)The computation is well explain in the appendix of [1].

Lemma 3.3.14. With the notation of Proposition3.3.13, let Y =(Y(n−1,0), Y(n,0), Y(n−2)

)if n is even, and Y =

(Y(n−1,0), Y(n−1,2), Y(n,1)

)if n is odd, then

MY t = (q2n−1 − 1,−1,−1,−1)t

where

(1) If n is even and p = 2,

M =

1 1 1−1qn−1 1 −1

qn−1qn−2qn−1+1

(qn−1)(qn−1−1)−1

qn−1−1−1qn−1

−1qn−1−1

−1qn−1−1

qn−1+1(qn−1)(qn−1−1)

(2) If n is odd and p = 2,

M =

1 1 11 −1

qn−1−1qn−1

−1qn−1

qn−2qn−1+1(qn−1)(qn−1−1)

−1qn−1

−1qn−1

qn+1(qn−1)(qn−1−1)

−1qn−1

(3) If n is even and p 6= 2,

M =

1 1 1−1qn−1

qn/2−1qn−1

−qn/2−1

qn−1qn−2qn−1+1

(qn−1)(qn−1−1)qn−1−qn/2−1(q−1)+1

(qn−1)(qn−1−1)qn−1−qn/2−1

(qn−1)(qn−1−1)−1

qn−1−1qn−1−qn/2−1(q+1)+1

(qn−1)(qn−1−1)qn−1+qn/2−1(q+1)+1

(qn−1)(qn−1−1)

(4) If n is odd and p 6= 2,

M =

1 1 1

qn+12 +1qn−1

−qn/2−1qn−1

−1qn−1

qn−2qn−1+1−q(n−1)/2(q−1)(qn−1)(qn−1−1)

qn−2qn−1+1+q(n−1)/2(q−1)+1(qn−1)(qn−1−1)

−1(qn−1)

qn+1−q(n−1)/2(q+1)(qn−1)(qn−1−1)

qn+1+q(n−1)/2(q−1)(qn−1)(qn−1−1)

−1(qn−1)

Proposition 3.3.15. The linear systems (S′) define above have no solution if n ≥ 3.

Proof. We solve the first three equations using a computer algebra system to get theunique solution Y ∗, and then we evaluate the left hand side of the last equation atY ∗, and observe that it cannot be equal to −1 unless n = 1, 2.

(i) If n is even and p = 2,

Y ∗ =

((qn − 1)(qn−1 − 1)

q − 1, qn−1 − 1,

(qn − 1)(qn − 2qn−1 + 1)

q − 1

)


and the left hand side of the last evaluated at Y ∗ is equal to

−1− 2qn(qn−2 − 1)

(q − 1)(qn−1 − 1).

(ii) If n is odd and p = 2,

Y ∗ =

(qn−1 − 1,

q(qn−1 − 1)2

q − 1, (q2n−1 − 1)− (qn−1 − 1− 1)

q − 1

)and the left hand side of the last evaluated at Y ∗ is equal to

−1− 2qn(qn−1 − 1)

(q − 1)(qn − 1).

(iii) If n is even and p 6= 2,

Y ∗ = ((qn − 1)(qn−1 − 1)

q − 1,(q2n−1 − 1)

2+

(qn/2(q − 1)− 1)(qn−1 − 1)

2(q − 1),

(q2n−1 − 1)

2+

(−qn/2(q − 1)− 1)(qn−1 − 1)

2(q − 1))


−1− 2qn(qn−2 − 1)

(q − 1)(qn − 1).

(iv) If n is odd and p 6= 2,

Y ∗ = ((qn − 1)(q(n−1)/2 + 1)(q(n+1)/2 − 1)

2(q − 1),(q(n−1) − 1)(q(n−1)/2 − 1)(q(n+1)/2 − 1)

2(q − 1),

(q2n−1 − 1)− (qn−1 − 1)

q − 1)


−1 + 2qn(qn−1 − 1)

(q − 1)(qn − 1).

Since the system does not have a solution, we can conclude there is no subspaceof Qn of dimension equals to (n − 1)(n − 2)/2. Therefore, the result of the proof ofthe Theorem 3.1.10 follows. And then, we have the following consequence.

Corollary 3.3.16. Let F be a finite field and Let L/F be an extension and suppose F alge-braic closed in L. Let S be a subspace of L with finite dimension greater or equals than two.And let A be a 2−atom of S, then dimA = 2.

Theorem 3.3.17. Let F be finite field and Let L/F be an extension and suppose F algebraicclosed in L. Let S, T be a subspaces of L such that 2 ≤ dimS, dimT and dimST ≤ dimL−2. If dimST = dimS+dimT −1 then, there are bases of S and of T respectively of the form{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1

}for some g, g′, a ∈ L.



Proof. The Corollary 3.3.16 implies that the 2−atoms has a dimension 2 so the resultwill follow from the Lemma2.1.2 and Lemma2.1.3 the result holds.

In this chapter, we were supposed to come out the link between Vosper’s theoremand quadratic forms and answer the question in the case of finite field. In order tosolve the problem, we were supposed to show the 2−atoms has only dimension 2.To deal with this problem, we have used the transformation of our problem into aproblem in the space of quadratic forms Qn. In the next chapter, we are going toinvestigate the case of transcendental fields for sure we will use other materials.

27

Chapter 4

Vosper’s theorem for fields withvaluations

In this chapter, we will first start by motivate why is important think about the tran-scendental extension, and we will prove the main result. This result, will generalizethe case where the base field is algebraically closed as in [1]. In the second part, wewill define the notion of Ci as in [7] field and some properties and then, we will de-duce from those properties two examples of C1 and then, apply the main result tohave the proof of the Vosper’s theorem for some examples C1 that we have found.

4.1 Motivation and Main result

In this section, we consider two finite-dimensional subspaces S and T such thatdimST = dimS + dimT − 1, but they live in an infinite dimensional extension L/F,with no element of L\F algebraic over F. Someone can ask a natural question, whendimS, dimT ≥ 2, can we conclude that S and T have bases in geometric progressionas in Chapter3? Our motivation to solve this question, come from the classical addi-tive setting in [12]: it is much easier to prove in the set Z of integer than Z/pZ, thatif |S|, |T | ≥ 2 and |S + T | = |S|+ |T | − 1, then S and T must be arithmetic progres-sions. This leads us to think that the case of infinite extension should be easier thatthe finite case. It turns out that, the answer of the question is yes but only in part.Since we know that, the transcendental extensions have valuation rings which allowus to transfer the structure of spaces with small products to sets with small sumsetsin totally ordered abelian groups. We will start by the following Lemma, which isvery important to establish the main result of this part.

Lemma 4.1.1. Let F be a field. Let L/F be a non-trivial extension and suppose F algebraicclosed in L. Let v be a valuation function on L such that its residue field Lv of L is containedin F and let T be a subspaces of L of finite dimension n. Then, there exist a basis (e1, · · · , en)of T such that v(e1) > v(e2) > · · · > v(en).

Proof. Let (e1, · · · , en) be a basis of T. If v(en−1) > v(en) there are nothing to dowe consider en−1 and en. If not, we consider v(en) = v(en − 1) this implies thatv(e−1n .en−1) = 0. Hence e−1n .en−1 ∈ Ov \ mv, where Ov is a valuation ring and mv

its maximal ideal. Therefore, there is some λ ∈ Lv such that en−1 = λen + xen forx ∈ mv. Finally, we have

v(en−1 − λen) = v(xen) = v(x) + v(en) > v(en). (4.1)

Now, we can take en−1 = en−1 − λen and we apply the same algorithm on ej andej−1 for j = n−1, n−2, · · · 1.At the end we have a new basis satisfying the conditionof the lemma.

28 Chapter 4. Vosper’s theorem for fields with valuations

Theorem 4.1.2 (Main Result). Let F field and Let L/F be a non-trivial extension andsuppose F algebraic closed in L. Let S, T be a subspaces of L such that 2 ≤ dimS, dimTand dimST ≤ dimL − 2 and dimST = dimS + dimT − 1. If L has a valuation vsuch that its residue field is contained in F then, S and T are respectively of the form{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1


Proof. As we have done in the previous chapter, it suffices to show that there exista space A of dimension 2 such that dimSA = dimS + 1. The result will thereforefollow by induction if we can show that when dimT > 2, there exists a subspace ofT ′ such that dimT ′ = dimT − 1 and dimST ′ ≤ dimS + dimT ′ − 1. One can supposeL generated by S and T, whence the transcendental degree is finite. Indeed, S andT are contained in the sub-field of L, which has finite transcendental degree. Since,Lv is contained in F , we can apply the previous Lemma 4.1.1, we may choose a basis(τ1, · · · , τt) of T such that w(τ1) > w(τ2) > · · · > w(τt). Setting T ′ to be the subspacegenerated by τ1, · · · , τt−1, we see that element of minimum valuation cannot exist inST ′. Therefore,

dimST ′ < dimST = dimS + dimT − 1 it follows that,dimST ′ ≤ dimS + dimT − 2 = dimS + dimT ′ − 1.

Now, using this result which is the main result of this part we will answer thequestion on Vosper’s theorem for some fields. We are going to identify some exten-sions L/F with valuation v such that Lv ⊂ F.

4.2 Applications

First we will solve the case where F is algebraically closed and L/F be a non-trivialextension. The second point consist to answer the same question for some C1 field.

Theorem 4.2.1. Let F be algebraically closed field and Let L/F be a non-trivial extensionand suppose F algebraic closed in L. Let S, T be a subspaces of L such that 2 ≤ dimS, dimTand dimST ≤ dimL − 2. If dimST = dimS + dimT − 1 then, there are bases of S andof T respectively of the form

{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1

}for some

g, g′, a ∈ L.

Proof. Without lost of generality, one can suppose L generated by S and T, whencethe transcendental degree is finite. SettingX = (x1, x2, · · · , xd) andα = (α1, α2, · · · , αd),onF (x1, x2, · · · , xd) we can define a valuation as follow v : F1 := F (x1, x2, · · · , xd)→Zd with Zd endowed with the lexicographic order such that v(xα1

1 , xα22 , · · · , xαd

d ) =(α1, α2, · · · , αd). and v(

∑aαX

α) = min{β : aβ 6= 0}. Since [F1 : L] is finite we canextend the valuation of F1 to get a valuationw on L (see [13]). Since, the residue fieldof F1 is F which is algebraically closed, it follows that, the residue field of L shouldbe F. From Theorem 4.1.2 we have the result.

Before continuing we are going to define what is calledCi field. All the followingdefinitions that we are going to give can be found in [7].

Definition 4.2.2. A field F is said to be C0 if every form in F in n variables and degree d,with n = d has a non trivial zero in F.

Remark 4.2.3. The definition above is another definition of algebraic closure. We know that,the algebraic closure of F is an algebraic extension of F that is algebraically closed.

4.3. Another idea when L has no F− valued place 29

Definition 4.2.4. A field F is said to be Ci with i ≥ 1, if every form in F in n variables anddegree d, with n > di has a non trivial zero in F.

Theorem 4.2.5. Let F be a Ci then, the rational function field and the formal powers seriesfield K((t)) are Ci+1

Proof. The proof can be seen in [7].

Corollary 4.2.6. If F is algebraically closed then, F (t) and F ((t)) are C1

In the next part, we will consider k be a algebraically closed field, F = k((t))and L be an extension of F. Since F is a complete discrete valuation field with al-gebraically closed residue field its extension L has a valuation and its residue fieldshould be again k. and by the Theorem 4.1.2 we have the existence of geometricalbasis for some S, T finite subspace of L satisfying some conditions. We resume whatwe have said by the following theorem.

Theorem 4.2.7. Let k be algebraically closed field, F = k((t)) and Let L/F be a non-trivialextension. Let S, T be a subspaces ofL such that 2 ≤ dimS, dimT and dimST ≤ dimL−2.If dimST = dimS + dimT − 1, then there are bases of S and of T respectively of the form{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1


4.3 Another idea when L has no F− valued place

What happen when F is not algebraically closed? It turns out that L has no F−valueplace, in which case we can only guarantee the existence of places with values in thealgebraic extension of F. If we consider F = Fq the result should be obtained if wehave already have the Vosper’s Theorem for finite extensions.

Theorem 4.3.1. Let F be a finite field Fq and Let L/F be an infinite extension such that noelement of L\F is algebraic over F. Let S, T be a subspaces of L such that 2 ≤ dimS, dimTand dimST ≤ dimL − 2. If dimST = dimS + dimT − 1, then there are bases of S andof T respectively of the form

{g, ga, · · · , gadimS−1

}and

{g, g′a, · · · , g′adimT−1

}for some

g, g′, a ∈ L.

Proof. As we have proved in Theorem4.2.1, we need to exhibit a subsapce T ′ of codi-mension 1 in T such that dimST ′ ≤ dimS + dimT ′ − 1. By the Lang-Weil estimationof number of rational points of an algebraic variety see [8], for any sufficiently largem there exists a place of Lwith value in Fmq a finite extension of F. From this we havea valuation v on L into a ordered abelian group. Let denote by O the valuation ringin L and P its maximal ideal, we have this isomorphismO/P ' Fmq . We choose m tobe prime and such that that m ≥ dimST + 2. Without loss of generality, translatingT if need be, we may suppose that the minimum valuation of element of T is O. LetTP = T ∩ P and decompose T as T = T0 ⊕ TP where T0 is any subspace of T indirect sum with TP . Note that since 0 is the minimum valuation in T, we have thatT0 6= {0} and all non-zero elements of T0 are of valuation 0. Let S = S0 ⊕ SP be asimilar decomposition.

We have for free ST ⊃ S0T0 + SPTP . Let denote by E be a maximal subspace ofS0T0 all of whose non-zero elements have valuation zero. We therefore have

ST ⊃ E ⊕ SPTP .

We make the remark that if s ∈ S0 and if τ is an element of TP of minimum valuation,then the valuation of sτ equals the valuation of τ and does not belong to the set of

30 Chapter 4. Vosper’s theorem for fields with valuations

valuations of the space E + SPTP . Therefore sτ /∈ E + SPTP and dimST > dimE +dimSPTP this means that

dimST ≥ dimE + dimSPTP + 1. (4.2)

The map π : O → O/P is injective on E and we have

dimE = dimπ(E) = dimπ(S0)π(T0).

By Cauchy-Davenport applied to the extensionO/P of F we have dimE ≥ dimS0 +dimT0−1. From the Cauchy-Davenport inequality applied inLwe have that dimSPTP ≥dimSP +dimTP −1. Since dimST = dimS+dimT −1 by the hypothesis of the Theo-rem, the inequality (4.2) implies that dimS+dimT − 1−dimSPTP − 1 ≥ dimE then,dimS + dimT − 1− dimSP − dimTP + 1. Therefore, we have

dimS0 + dimT0 − 1 ≥ dimE.

It turns out that dimE = dimS0 + dimT0 − 1. Now, if we suppose dimS0 ≥ 2 anddimT0 ≥ 2, because we have the extension Fmq sufficiently large, Theorem 3.3.17applies in Fmq /Fq and there exist a subspace T ′0 of T0 of codimension 1 such thatdimπ(S0)π(T ′0) < dimπ(S0)π(T0) and T ′ = T ′0 ⊕ TP is the subspace of T of codimen-sion 1 that we are looking for. If either dimS0 = 1 or dimT0 = 1, then T ′ = T ′0 + TPfor any subspace T ′0 of codimension 1 of T0 again yields the required subspace ofT.

31

Chapter 5

Conclusion

The aim of this work, was to study some analogue results of additive theory. Espe-cially the transposition of Vosper’s theorem in the case of field extension. We havestarted by setting the scene, to clarify exactly what we are going to do in the intro-duction.

In the Chapter2, we studied the case where F is algebraically closed in L a finiteextension such that [L : F ] − 2 is prime. In this case, we have shown that if T, S aretwo F− subspaces of Lwith dimension greater or equal to two the Vosper’s theoremholds.

Secondly, in Chapter 3, we have firstly given the motivation to study some fieldslike algebraically closed field and finite field. After, we gave the connection betweenquadratic form and Vosper’s theorem, and we redefine our problem. In the last partof this chapter, we have shown that in the case of finite finite field the result holds.

The third part of this project, was to understand how the same the Vosper’s theo-rem could work in the the case of fields with valuations. After a brief explanation ofour ideas, we have shown why in the field extension L/F with residue field Lv ⊂ Fthe Vosper’s theorem works. And we identify some field satisfying this condition.However, when F is not algebraically closed we can only guarantee the existence ofplace with values in an algebraic extension of F. In this type of situation, we havebrought the answer only for the case of finite field.

So far, we don’t have complete answer of this problem. Only a small piece ofthe classical results of the additive theory have been obtained, and many questionsare wide open. For instance, some one could ask himself how can we generalize theresult in the case of field with valuation?

32

Bibliography

[1] Christine Bachoc, Oriol Serra, and Gilles Zemor. “An analogue of Vosper’sTheorem for Extension Fields”. In: arXiv preprint arXiv:1501.00602 (2015).

[2] Christine Bachoc, Oriol Serra, and Gilles Zémor. “Revisiting Kneser’s Theoremfor Field Extensions”. In: arXiv preprint arXiv:1510.01354 (2015).

[3] Shalom Eliahou and Cédric Lecouvey. “On linear versions of some additiontheorems”. In: Linear and multilinear algebra 57.8 (2009), pp. 759–775.

[4] Yahya O Hamidoune. “An isoperimetric method in additive theory”. In: Jour-nal of Algebra 179.2 (1996), pp. 622–630.

[5] Yahya O Hamidoune. “Some additive applications of the isoperimetric ap-proach”. In: Annales de l’institut Fourier. Vol. 58. 6. 2008, pp. 2007–2036.

[6] Yahya Ould Hamidoune. “Some results in additive number theory I: The crit-ical pair theory”. eng. In: Acta Arithmetica 96.2 (2000), pp. 97–119. URL: http://eudml.org/doc/278989.

[7] Serge Lang. “On quasi algebraic closure”. In: Annals of Mathematics (1952),pp. 373–390.

[8] Serge Lang and André Weil. “Number of points of varieties in finite fields”.In: American Journal of Mathematics 76.4 (1954), pp. 819–827.

[9] H.B. Mann. Addition theorems: the addition theorems of group theory and numbertheory. Interscience tracts in pure and applied mathematics. Interscience Pub-lishers, 1965. URL: https://books.google.fr/books?id=vw7vAAAAMAAJ.

[10] Kai-Uwe Schmidt. “Symmetric bilinear forms over finite fields of even char-acteristic”. In: Journal of Combinatorial Theory, Series A 117.8 (2010), pp. 1011–1026.

[11] Kai-Uwe Schmidt. “Symmetric bilinear forms over finite fields with appli-cations to coding theory”. In: Journal of Algebraic Combinatorics 42.2 (2015),pp. 635–670.

[12] TERENCE Tao. “Lecture notes 2 for 254a”. In: Univ. Calif. Los Angeles, http://www.math. ucla. edu/ tao/254a 1 (2010).

[13] Michel Vaquié. “Valuations and local uniformization”. In: Advanced Studies inPure Mathematics 4 (2008), p. 200.

[14] Alan Gordon Vosper. “The critical pairs of subsets of a group of prime order”.In: Journal of the London Mathematical Society 1.2 (1956), pp. 200–205.

http://eudml.org/doc/278989

http://eudml.org/doc/278989

https://books.google.fr/books?id=vw7vAAAAMAAJ

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Linear analogue of additive theorems · the additive theory have been obtained, and many questions are wide open. The goal of this master thesis is to understand what has been established